Answer: throw the five-trey and keep your trump

Throw the five-trey.

Start with the trump. There are seven aces. The double-ace, ace-blank, deuce-ace and trey-ace went in the first trick. The six-ace and four-ace went in the second. That is six. The seventh is the five-ace in your hand. Nobody else has a trump.

That means North's double-six cannot lose. It is the highest six, and nobody can trump it. If you put your five-ace on it, you spend your last trump on a trick your side already had.

So you throw something. The question is which domino is going to hurt you later. Look at the five-trey. The double-five and the five-blank are both still out, and that is 15 points. The double-five is the highest five, so your five-trey can never beat it. If you hang on to the five-trey, sooner or later it has to be played, either because you are on lead with nothing else or because fives are led and you have to follow suit. Either way, it walks into the double-five, and the count goes with it.

Throw it now and you have no fives at all. Having no dominoes left in a suit is called being void in it. Once you are void in fives, a five lead does not force you to do anything, and your five-ace can trump it.

Here is how the real hand went after you threw the five-trey. North led the four-trey. East's only four was the six-four, so East had to follow with it. You won with the double-four: 11 points. Then you led the five-ace. Nobody had a trump to follow with, and West let the double-five go under it: 16 more. The last trick went to West, with the five-blank in it. Your side finished with 36 and made the 31.

Throw the trey-blank instead and you are still holding the five-trey at the end. On the last trick it goes down, West's double-five takes it, and the five-blank falls on it too. That is 16 points to them. Your side ends with 26 and is set. Trump North's double-six and it is worse, 20.

We checked this with a small program. It tried every way the dominoes you cannot see could be split that fits the play so far, 4,200 of them, and played the rest of the hand out the best way for both sides each time. Throwing the five-trey did best on average. It was the best play, or tied for best, in 4,144 of the 4,200. In the other 56 a different throw did a little better, but nothing you can see from your seat tells you that you are in one of those.

The habit: do not trump a trick your partner has already won. And when you have to throw something, throw the domino that can only lose to the count.